Theorem ofcc 31974

Description: Left operation by a constant on a mixed operation with a constant. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcc.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcc.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofcc.3	⊢ (𝜑 → 𝐶 ∈ 𝑋)

Assertion

Ref	Expression
ofcc	⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofcc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcc.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		fnconstg 6646	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
4		ofcc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		ofcc.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
6		fvconst2g 7059	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
7	1, 6	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑥) = 𝐵)
8	3, 4, 5, 7	ofcfval 31966	. 2 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
9		fconstmpt 5640	. 2 ⊢ (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))
10	8, 9	eqtr4di 2797	1 ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f/c 𝑅𝐶) = (𝐴 × {(𝐵𝑅𝐶)}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4558   ↦ cmpt 5153   × cxp 5578   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   ∘_f/c cofc 31963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-ofc 31964

This theorem is referenced by: (None)